Finite Group Databasebeta

← #41012473

Subgroups — #41012473

Order 27008.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27213856C211yesmaximal, characteristic, complemented
2#27213519C211yesmaximal, characteristic, complemented
3#27213346C211yesmaximal, characteristic, complemented
4#27213342C211yesmaximal, characteristic
5#27212894C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7Q8⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—22no—
9C2×#16809949—21no—
10C2×D1688—21no—
11#20269977C2211yescharacteristic
12#20269974C2211yescharacteristic
13#20269969—21no—
14#20269955—22no—
15#20269953—21no—
16C4×D844—21no—
17C4×D844C2211yescharacteristic
18C4⋊C4×D211—21no—
19C4⋊(C211⋊C8)C2211yescharacteristic
20#20269899—21no—
21#20269897—21no—
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422D412yes—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844D411yescharacteristic
30C2×C4×D211—22no—
31C2×C4×D211—21no—
32#16809951—44no—
33#16809949—42no—
34D1688—42no—
35C2×C211⋊C8—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211D411yescharacteristic
39Dic211⋊C4—22no—
40Dic211⋊C4—22no—
41(C2×D211)⋊C4—22no—
42(C2×D211)⋊C4—21no—
43C2×Q8×C211D411yescharacteristic
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51C22×D211—41no—
52Dic422—42no—
53D844—21no—
54D844—42no—
55D844—41no—
56D844—41no—
57D844—41no—
58D844—41no—
59D844SD1612yescomplemented
60C2×Dic211—21no—
61C2×Dic211—22no—
62C4×D211—41no—
63C4×D211—41no—
64C4×D211—82no—
65C211⋊C8—41no—
66Q8×C211—21no—
67Q8×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—22no—
74C8×C211—41no—
75C22×C211#12011yescharacteristic
76C2×D211—22no—
77C2×D211—41no—
78C2×D211—41no—
79C2×D211—81no—
80C2×D211—81no—
81C2×D211—81no—
82C4×C211—21no—
83C4×C211—41no—
84C4×C211—41no—
85C4×C211C2×SD1611yescharacteristic
86C4×C211#13511yescharacteristic
87C4×C211AGL(1,ℤ/8)11yescharacteristic
88C4×C211#13711yescharacteristic
89Dic211—41no—
90Dic211—42no—
91C2×C211#44811yescharacteristic
92C2×C211#44911yescharacteristic
93C2×C211#45311yescharacteristic
94D211—42no—
95D211—161no—
96C211#162211yescharacteristic, complemented, Sylow
97#1622—2111nomaximal, Sylow
98#554—2111no—
99#492—2111no—
100#460—2111no—
101#459—2111no—
102#333—2111no—
103Q8⋊C8D21111yescharacteristic, complemented
104D4⋊C8—2111no—
105C2×SD16—4222no—
106C2×SD16—4221no—
107C2×D8—4221no—
108C4⋊Q8—2111no—
109#127—2111no—
110#125—4221no—
111C22⋊Q8—4222no—
112C22⋊Q8—4221no—
113C4×Q8C2×D21111yescharacteristic
114C4×D4—4221no—
115C4×D4—2111no—
116C2×C4⋊C4—4221no—
117C4⋊C8—2111no—
118C4⋊C8C2×D21111yescharacteristic
119#102—4221no—
120#102—4221no—
121C4×C8C2×D21111yescharacteristic
122C2×Q8—2112no—
123C2×Q8D84411yescharacteristic
124C2×D4—4221no—
125C2×D4—4221no—
126C2×D4—4221no—
127C2×D4—2111no—
128C22×C4—4222no—
129C22×C4—4221no—
130SD16—8444no—
131SD16—8442no—
132D8—8442no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—4221no—
136C4⋊C4—21no—
137C4⋊C4—4222no—
138C4⋊C4—4222no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—2111no—
142C4⋊C4D84411yescharacteristic
143C22⋊C4—4222no—
144C22⋊C4—4221no—
145C42—21no—
146C42C22×D21111yescharacteristic
147C23—8441no—
148C23—4221no—
149Q8—21no—
150Q8—21no—
151Q8—8442no—
152D4—8442no—
153D4—8441no—
154D4—8441no—
155D4—8441no—
156D4—8441no—
157D4—4221no—
158D4—2112no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—8441no—
162C2×C4—8441no—
163C2×C4—16882no—
164C2×C4—4221no—
165C2×C4—4222no—
166C2×C4D4×D21111yescharacteristic
167C2×C4D4×D21111yescharacteristic
168C2×C4C2×D84411yescharacteristic
169C8—22no—
170C8—41no—
171C8—8441no—
172C22—8441no—
173C22—8441no—
174C22—16881no—
175C22—16881no—
176C22—16881no—
177C22—4222no—
178C22#2026994711yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—8442no—
183C4—8441no—
184C4SD16×D21111yescharacteristic
185C4#2026999911yescharacteristic
186C4#2027000511yescharacteristic
187C4#2027000911yescharacteristic
188C2—33761no—
189C2—8442no—
190C2#2721328611yescharacteristic
191C2#2721329311yescharacteristic
192C2#2721331511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingQ8⋊C8×C211C2
socleC22×C211#120
prime radical#27213342C2
hypercenterQ8⋊C8D211
hypocenterC211#1622
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #27213856C2semidirect
2. #27213519C2semidirect
3. #27213346C2semidirect
5. #27212894C2semidirect
6. #27212875C2semidirect
7. Q8⋊C8×C211C2semidirect
59. D844SD16semidirect
96. C211#1622semidirect
103. Q8⋊C8D211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#272138562
#272135192
#272133462
#272133422
#272128942
#272128752
Q8⋊C8×C2112
#1622211